Daniel Peterseim

Localization of elliptic multiscale problems

This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size

Oversampling for the multiscale finite element method

H

Computation of eigenvalues by numerical upscaling

, patches of diameter

Convergence of a Discontinuous Galerkin Multiscale Method

H\log (1/H)

A localized orthogonal decomposition method for semi-linear elliptic problems

are sufficient to preserve a linear rate of convergence in

Multiscale Partition of Unity

H

Analysis-suitable adaptive T-mesh refinement with linear complexity

without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. - See more at: http://www.ams.org/journals/mcom/2014-83-290/S0025-5718-2014-02868-8/#sthash.z2CCFXIg.dpuf

Optimal adaptive nonconforming FEM for the Stokes problem

This paper reviews standard oversampling strategies as performed in the multiscale finite element method (MsFEM). Common to those approaches is that the oversampling is performed in the full space ...

Two-Level Discretization Techniques For Ground State Computations Of Bose-Einstein Condensates

We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we consider benchmark multi-scale eigenvalue problems in reservoir modeling and material science. We compute a low-dimensional generalized (possibly mesh free) finite element space that preserves the lowermost eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of